Therefore we may naturally identify XX as a simplicial set equipped with a subset of X1X_1 that contains all degenerate 1-cells.

Moreover, a morphism of separated preseheaves on Δ+\Delta^+ is by definition just a natural transformation between them, which means it is under this interpretation precisely a morphism of simplicial sets that respects the marked 1-simplices.

Cartesian closure

Lemma

Proof

This is an immediate consequence of the above observation that sSet+sSet^+ is a quasitopos. But it is useful to spell out the Cartesian closure in detail.

By the general logic of the closed monoidal structure on presheaves we have that PSh(Δ+)PSh(\Delta^+) is cartesian closed. It remains to check that if X,Y∈PSh(Δ+)X,Y \in PSh(\Delta^+) are marked simplicial sets in that X1+→X1X_{1^+} \to X_1 is a monomorphism and similarly for YY, that then also YXY^X has this property.

Now, by construction, every non-identity morphism U→[1+]U \to [1^+] in Δ+\Delta^+ factors through U→[1]U \to [1], which implies that if the components of p*η1p^* \eta_1 and p*η2p^* \eta_2 coincide on U≠[1+]U \neq [1^+], then already the components of η1\eta_1 and η2\eta_2 on UU coincided. By assumption on XX the values of η1\eta_1 and η2\eta_2 on U=[1+]U = [1^+] are already fixed, due to the inclusion X1+×[1+]1+↪X1×[1+]1X_{1^+} \times [1^+]_{1^+} \hookrightarrow X_{1} \times [1^+]_{1}. Hence p*p^* is injective, and so YXY^X formed in PSh(Δ+)PSh(\Delta^+) is itself a marked simplicial set.

The nn-simplices of the internal hom YXY^X are simplicial maps X×Δ[n]→YX \times \Delta[n] \rightarrow Y such that when you restrict X1×Δ[n]1→Y1X_1 \times \Delta[n]_1 \rightarrow Y_1 to E×Δ[n]0E \times \Delta[n]_0 (where EE is the set of marked edges of XX), this morphism factors through the marked edges of YY.

The marked edges of YXY^X are those simplicial maps X×Δ[1]→YX \times \Delta[1] \rightarrow Y such that the restriction of X1×Δ[1]1→Y1X_1 \times \Delta[1]_1 \rightarrow Y_1 to E×Δ[1]1E \times \Delta[1]_1 factors though the marked edges of YY. In the presence of the previous condition, this says that when you apply the homotopy X×Δ[1]→YX \times \Delta[1] \rightarrow Y to a marked edge of XX paired with the identity at [1][1], the result should be marked.

Definition

We generalize all this notation from sSet+sSet^+ to the overcategorysSet+/S:=sSet+/(S♯)sSet^+/S := sSet^+/(S^\sharp) for any given (plain) simplicial set SS, by declaring

MapS♭(X,Y)⊂Map♭(X,Y)
Map_S^\flat(X,Y) \subset Map^\flat(X,Y)

and

MapS♯(X,Y)⊂Map♯(X,Y)
Map_S^\sharp(X,Y) \subset Map^\sharp(X,Y)

to be the subcomplexes spanned by the cells that respect that map to the base SS.

Proof

The nn-cells of MapS♭(X,Y♮)Map_S^\flat(X,Y^\natural) are morphisms X×Δ[n]♭→Y♮X \times \Delta[n]^\flat \to Y^\natural over SS. This means that for fixed x∈X0x \in X_0, Δ[n]\Delta[n] maps into a fiber of Y→SY\to S. But fibers of Cartesian fibrations are fibers of inner fibrations, hence are quasi-categories.

Similarly, the nn-cells of MapS♯(X,Y♮)Map_S^\sharp(X,Y^\natural) are morphisms X×Δ[n]♯→Y♮X \times \Delta[n]^\sharp \to Y^\natural over SS. Again for fixed x∈X0x \in X_0, Δ[n]\Delta[n] maps into a fiber of Y→SY\to S, but now only hitting Cartesian edges there. But (as discussed at Cartesian morphism), an edge over a point is Cartesian precisely if it is an equivalence.

Notice that trivially every object in this model structure is cofibrant. The following proposition shows that the above model structure indeed presents the (∞,1)(\infty,1)-category CartFib(S)CartFib(S) of Cartesian fibrations.

Proposition

An object p:X→Sp : X \to S in sSet+/SsSet^+/S is fibrant with respect to the above model structure precisely if it is isomorphic to an object of the form Y♮Y^\natural, for Y→SY \to S a Cartesian fibration in sSet.

Proof

In particular, the fibrant objects of sSet+≅sSet+/*sSet^+ \cong sSet^+/* are precisely the quasicategories in which the marked edges are precisely the equivalences. Note that the Cartesian model structure on sSet+/SsSet^+/S is not the model structure on an over category induced on sSet+/SsSet^+/S from the Cartesian model structure on sSet+sSet^+!

Definition/Proposition

(coCartesian model structure on sSet+/SsSet^+/S)

There is another such model structure, with Cartesian fibrations replaced everywhere by coCartesian fibrations.

Marked anodyne morphisms

A class of morphisms with left lifting property again some class of fibrations is usually called anodyne . For instance a left/right/inner anodyne morphism of simplicial sets is one that has the left lifting property against all left/right fibrations or inner fibrations, respectively.

The class of marked anodyne morphisms in sSet+sSet^+ as defined in the following is something that comes close to having the left lifting property against all Cartesian fibrations. It does not quite, but is still useful for various purposes.

Proof

Remark

Thus, if (X,EX)→(S,ES)(X, E_X) \to (S,E_S) is a morphism in sSet+sSet^+ with RLP against marked anodyne morphisms, then its underlying morphism X→SX\to S in sSetsSet is almost a Cartesian fibration: it may fail to be such only due to missing markings in ESE_S.

However, if all morphisms in SS are marked, then (X,EX)→S♯(X,E_X) \to S^\sharp has the RLP against marked anodyne morphisms precisely when the underlying morphism X→SX\to S is a Cartesian fibration and exactly the Cartesian morphisms are marked in XX, (X,EX)=X♮(X,E_X) = X^\natural — in other words, precisely if it is a fibrant object in the model structure on sSet+/SsSet^+/S.